the use of Cartesian product I am currently studying in secondery level. I read cartesian product the other day and I found it absolutely bizarre. In my text book, there is this "order pair" which I understood fairly well and then there is cartesian product in which we multiply two sets. I don't understand the concept behind it. What is its application? How will it come to use in higher level study?
 A: The basic point of set theory in mathematics is that it allows us to wrap up properties and treat them as if they were things.
Without sets we can say things like

$2.5$ is a rational number.
$\pi$ is not a rational number.
$36$ is a perfect square.
$(0,42)$ is an ordered pair of two whole numbers.

The wordings "is a rational number", "is a perfect square", "is an ordered pair of two whole numbers" denote properties. They are nice for talking about particular things that have or don't have the property. But our language doesn't allow us to speak about, for example, an expression whose value is a property, or a function that takes a property as input or things like that.
This is where sets come in. A set is a way to "box up" a property such that it becomes a "thing" that we can speak about together with other similar things. To a large extent, a property and a set are different ways of thinking about the same concept. If we have a property, say "is a thingamajig" we can always (with some technical exceptions that you don't need to worry about at this level) create a set from it: $$A=\{x\mid x\text{ is a thingamajig}\}$$
On the other hand, whenever we have a name or an expression for a set, this gives us a property: "is a member of such-and-such set".
Because sets are "things", they can have (or not have) properties themselves -- such as the property "is a set of real numbers (but possibly not all the real numbers)". Boxing up these properties, we get sets of sets and so forth.
Now the cartesian product $A\times B$ is neither more nor less than the set corresponding to the property "is an ordered pair of something from $A$ and something from $B$".
Thus, whenever we want to speak about ordered pairs at all, it may suddenly become useful to speak about the property of being an ordered pair. And then it immediately becomes useful to speak about the boxed-up property that corresponds to it. In fact so useful that the common way to say "is an ordered pair of something from $A$ and something from $B$" is "is in $A\times B$". This is shorter and more concise.
So cartesian products are used, or potentially used, everywhere we use ordered pairs in the first place.
A: The Cartesian product of $2$ sets $A$ and $B$ is just the set of all ordered pairs $(a,b)$ where $a \in A$ and $b \in B$. You can think of it as creating a set of from 2 other sets. For example $A = B = \mathbb{R} => A\times B = \mathbb{R}^{2}$. Put two real number lines perpendicular to each other and you get the xy-plane. Another example is $A=\{6,8\}, B=\{1,2,5\} => A \times B = \{(a,b)\}$ where a could be either 6 or 8 and b could be 1, 2 or 5. Thus an element of $A\times B$ could be $(6,2)$, but not $(8,7)$ since $7$ is not in $B$.
To answer: 'How will it come to use in higher level study?'
Some branches of mathematics use "The Axiom of Choice" which is equivalent to "the cartesian product of a collection of non-empty sets is non-empty". Said Axiom is used in measure theory to prove that there exists sets that cannot be "measured".
A: If you understand what ordered pairs are, then understanding the Cartesian product is not very difficult.
Given two sets $A$ and $B$ are, their Cartesian product, denoted by $A\times B$, is a new set whose elements are all the ordered pairs $(a,b)$ where $a\in A$ and $b\in B$.
For example if $A=\{0,1\}$ and $B=\{x\}$, then $A\times B=\{(0,x),(1,x)\}$. We took all the possible combinations.
What are its applications? Many. When $A$ and $B$ have additional structure of "the same type" we can often amalgamate these structures to endow $A\times B$ with a structure of that type as well, in a way which extends naturally the structure of $A$ and $B$.
This includes groups, rings, topological spaces, and many more.
In modern mathematics relations, and in particular functions, can be modeled as subsets of Cartesian products of sets, all of which make this notion incredibly important. 
A: also i just got to learn that we use set cartesian products in combinatorial problems .An example of this is a three course meal where there is an appetiser main course and desert.Lets say for example we have porridge,soup,lemonade and main course chicken,omlades,cooked potatoes,and desert icecream fruitpoodingand chocolate one can have soup,chicken,chocolate or soup,omlates,fruitpooding etc thus such is an example of application of set cartesian product
